Mechanism of self-organization in point vortex system

Abstract

A mechanism of the self-organization in an unbounded two-dimensional (2D) point vortex system is discussed. A kinetic equation for the system with positive and negative vortices is derived using the Klimontovich formalism. Similar to the Fokker-Planck collision term, the obtained collision term consists of a diffusion term and a drift term. It is revealed that the mechanism for the self-organization in the 2D point vortex system at negative absolute temperature is mainly provided by the drift term. Positive and negative vortices are driven toward opposite directions respectively by the drift term. As a result, well-known, two isolated clumps with positive and negative vortices, respectively, are formed as an equilibrium distribution. Regardless of the number of species of the vortices, either single- or double-sign, it is found that the collision term has following physically good properties: (i) When the system reaches a quasi-stationary state near the thermal equilibrium state with negative absolute temperature, the sign of $d ω/ d ψ$ is expected to be positive, where $ω$ is the vorticity and $ψ$ is the stream function. In this case, the diffusion term decreases the mean field energy, while the drift term increases it. As a whole, the total mean field energy is conserved. (ii) Similarly, the diffusion term increases the Boltzmann entropy, while the drift term decreases it. As a whole, the total entropy production rate is positive or zero ($H$ theorem), which ensures that the system relaxes to the global thermal equilibrium state characterized by the zero entropy production.